QERGD+: Quantum-Enhanced Recurrent Gradient Descent with Shor-Inspired Periodicity Priors for Stabilizing Dynamic Neural Network Training

This research introduces QERGD+, a hybrid quantum–classical algorithm for training recurrent neural networks (RNNs) and gated variants (GRUs, LSTMs) on long sequences without suffering from the vanishing/exploding gradient problem inherent to backpropagation through time (BPTT). The method replaces unstable recurrent-path gradients with exact, analytic parameter-shift derivatives computed on parameterized quantum circuits (PQCs) using Qiskit primitives. It augments training with a Shor-inspired, low-depth periodicity prior that biases learning toward stable cycles when periodic or quasi-periodic structure is present. The approach confines quantum resources to the recurrent bottleneck, remains NISQ-friendly, and maintains compatibility with large-scale classical architectures. Detailed pseudocode, Qiskit implementation patterns, complexity analysis, and empirical use-case guidance are provided.